I'm trying to solve exercise I.$3.17$ a) from Hartshorne's Algebraic Geometry, which states that every conic $X\subset\mathbb{P}^2$ is normal.
By definition $X$ is normal $\Leftrightarrow \mathcal{O}_{P,X}$ is integrally closed for every $P\in X$. Since this is a local property, I can think of $X$ as an affine conic.
I've tried to look at the specific example $X=Z(y-x^2)\subset\mathbb{A}^2$ to see if it helps.
Since $\overline{y}=\overline{x}^2\in A(X)$, then for any $f\in k[x,y]$ we get: $$f(\overline{x},\overline{y})=f(\overline{x},\overline{x}^2)=\prod_{i=1}^k(\overline{x}-a_i)$$
for some constants $a_i\in k$, since $k$ is algebraically closed.
This shows $A(X)$ is a UFD, and in particular integrally closed.
I'm sure this argument won't work for a general conic, so I'm stuck.
Any suggestions?